geometry chapter tests with key

Leah Aquino Cuenco

Text Box

GEOMETRY

Chapter TestCHAPTER

1

5A Chapter 1 Geometry English-Spanish Reviews

Use the diagram to name the figures.

1. Three collinear points

2. Four noncoplanar points

3. Two opposite rays

4. Two intersecting lines

5. The intersection of plane LMN and plane QLS

Find the length of the segment.

6. MPÆ

7. SMÆ

8. NRÆ

9. MRÆ

Find the measure of the angle.

10. ™DBE 11. ™FBC

12. ™ABF 13. ™DBA

14. Refer to the diagram for Exercises 10–13. Name an obtuse angle, an acute angle, a right angle, and two complementary angles.

15. Q is between P and R. PQ = 2w º 3, QR = 4 + w, and PR = 34.Find the value of w. Then find the lengths of PQ

Æand QR

Æ.

16. RTÆ

has endpoints R(º3, 8) and T(3, 6). Find the coordinates of the midpoint, S, of RT

Æ. Then use the Distance Formula to verify that RS = ST.

17. Use the diagram. If m™3 = 68°, find the measures of ™5 and ™4.

18. Suppose m™PQR = 130°. If QTÆ̆

bisects ™PQR, what is the measure of ™PQT?

The first five figures in a pattern are shown. Each square in the grid is

1 unit ª 1 unit.

19. Make a table that shows the distance around each figure at each stage.

20. Describe the pattern of the distances and use it to predict the distance around the figure at stage 20.

A center pivot irrigation system uses a fixed water supply to water

a circular region of a field. The radius of the watering system is 560

feet long. (Use π ≈ 3.14.)

21. If some workers walked around the circumference of the watered region,how far would they have to walk? Round to the nearest foot.

22. Find the area of the region watered. Round to the nearest square foot.

Copyright © McDougal Littell Inc.All rights reserved.

L

q

S

T N

MR

UP

X

1 2 3 4 5

536

4

S RM P N

826

A50�

CB

45�

DE

F

Chapter TestCHAPTER

2


State the postulate that shows that the statement is false.

1. Plane R contains only two points A and B.

2. Plane M and plane N are two distinct planes that intersect at exactly two distinct points.

3. Any three noncollinear points define at least three distinct planes.

4. Points A and B are two distinct points in plane Q. Line AB¯̆

does not intersect plane Q.

Find a counterexample that demonstrates that the converse of the statement is false.

5. If an angle measures 34°, then the angle is acute.

6. If the lengths of two segments are each 17 feet, then the segments are congruent.

7. If two angles measure 32° and 148°, then they are supplementary.

8. If you chose number 13, then you chose a prime number.

State what conclusions can be made if x = 5 and the given statement

is true.

9. If x > x º 2, then y = 14x. 10. If ºx < 2x < 11, then x = y º 12.

11. If |x| > ºx, then y = ºx. 12. If y = 4x, then z = 2x + y.

In Exercises 13–16, name the property used to make the conclusion.

13. If 13 = x, then x = 13. 14. If x = 3, then 5x = 15.

15. If x = y and y = 4, then x = 4. 16. If x + 3 = 17, then x = 14.

17. PROOF Write a two-column proof.

GIVEN � AXÆ

£ DX,Æ

XBÆ

£ XCÆ

PROVE � ACÆ

£ BDÆ

18. PLUMBING A plumber is replacing a small section of a leaky pipe. To find the length of new pipe that he will need, he first measures the leaky section of the old pipe with a steel tape measure, and then uses this measure to find the same length of new pipe. What property of segment congruence does this process illustrate? Use the wording of the property to explain how it is illustrated.

19. PACKAGING A tool and die company produces a part that is to be packed in triangular boxes. To maximize space and minimize cost, the boxes need to be designed to fit together in shipping cartons. If ™1 and ™2 have to be complementary, ™3 and ™4 have to be complementary, and m™2 = m™3, describe the relationship between ™1 and ™4.


TOOL &

DIE CO.

12 3

4

A B

D C

X

Chapter TestCHAPTER

3


In Exercises 1–6, identify the relationship between the angles in the

diagram at the right.

1. ™1 and ™2 2. ™1 and ™4

3. ™2 and ™3 4. ™1 and ™5

5. ™4 and ™2 6. ™5 and ™6

7. Write a flow proof. 8. If l ∞ m, which angles are

GIVEN � m™1 = m™3 = 37°, BAÆ̆

fi BCÆ̆ supplementary to ™1?

PROVE � m™2 = 16°

Use the given information and the diagram at

the right to determine which lines must be parallel.

9. ™1 £ ™2

10. ™3 and ™4 are right angles.

11. ™1 £ ™5; ™5 and ™7 are supplementary.

In Exercises 12 and 13, write an equation of the line described.

12. The line parallel to y = º�13�x + 5 and with a y-intercept of 1

13. The line perpendicular to y = º2x + 4 and that passes through the point (º1, 2)

14. Writing Describe a real-life object that has edges that are straight lines. Are any of the lines skew? If so, describe a pair.

15. A carpenter wants to cut two boards to fit snugly together. The carpenter’s squares are aligned along EF

Æ,

as shown. Are ABÆ

and CDÆ

parallel? State the theorem that justifies your answer.

16. Use the diagram to write a proof.

GIVEN � ™1 £ ™2, ™3 £ ™4

PROVE � n ∞ p

m

1

l

n

2 34

56 7

8

A

CB1

3

2


5 3

14

6

2

2ndF

m1l n

2

p

6 3q

5 74

2

m

1

l

n

p5

4

3

Chapter TestCHAPTER

4


In Exercises 1–6, identify all triangles in the figure that fit the given description.

1. isosceles 2. equilateral 3. scalene

4. acute 5. obtuse 6. right

7. In ¤ABC, the measure of ™A is 116°. The measure of ™B is three timesthe measure of ™C. Find m™B and m™C.

Decide whether it is possible to prove that the triangles are congruent. If it is possible,

tell which congruence postulate or theorem you would use. Explain your reasoning.

8. 9. 10.

11. 12. 13.

Find the value of x.

14. 15. 16.

PROOF Write a two-column proof or a paragraph proof.

17. GIVEN � BDÆ

£ ECÆ

, ACÆ

£ ADÆ

18. GIVEN � XYÆ

∞ WZÆ

, XZÆ

∞ WYÆ

PROVE � ABÆ

£ AEÆ

PROVE � ™X £ ™W

Place the figure in a coordinate plane and find the requested information.

19. A right triangle with leg lengths of 4 units and 20. A square with side length s and vertices at 7 units; find the length of the hypotenuse. (0, 0) and (s, s); find the coordinates of the

midpoint of a diagonal.

X Y

Z W

1 2

A

B C D E

x �

65�

x � 1

2x � 13x � 4

x �

70�

G H

M

K J

X

W Z Y

V

T

U

R

S

q

M

L

N

P

H

G

J

K

B

A CD F

E


q

P S R

Chapter TestCHAPTER

5


In Exercises 1–5, complete the statement with the word always,

sometimes, or never.

1. If P is the circumcenter of ¤RST, then PR, PS, and PT are � ?�� equal.

2. If BDÆ̆

bisects ™ABC, then ADÆ

and CDÆ

are � ?�� congruent.

3. The incenter of a triangle � ?�� lies outside the triangle.

4. The length of a median of a triangle is � ?�� equal to the length of a midsegment.

5. If AMÆ

is the altitude to side BCÆ

of ¤ABC, then AMÆ

is � ?�� shorter than ABÆ

.

In Exercises 6–10, use the diagram.

6. Find each length.

a. HC b. HB c. HE d. BC

7. Point H is the � ?�� of the triangle.

8. CGÆ

is a(n) � ?��, � ?��, � ?��, and � ?�� of ¤ABC.

9. EF = � ?�� and EFÆ

∞ � ?�� by the � ?�� Theorem.

10. Compare the measures of ™ACB and ™BAC. Justify your answer.

11. LANDSCAPE DESIGN You are designing a circular swimming pool for a triangular lawn surrounded by apartment buildings. You want the center of the pool to be equidistant from the three sidewalks. Explain how you can locate the center of the pool.

In Exercises 12–14, use the photo of the three-legged tripod.

12. As the legs of a tripod are spread apart, which theorem guarantees that the angles between each pair of legs get larger?

13. Each leg of a tripod can extend to a length of 5 feet. What is the maximum possible distance between the ends of two legs?

14. Let OAÆ

, OBÆ

, and OCÆ

represent the legs of a tripod. Draw and label a sketch. Suppose the legs are congruent and m™AOC > m™BOC. Compare the lengths of AC

Æand BC

Æ.

In Exercises 15 and 16, use the diagram at the right.

15. Write a two-column proof.

GIVEN � AC = BC

PROVE � BE < AE

16. Write an indirect proof.

GIVEN � AD ≠ AB

PROVE � m™D ≠ m™ABC


E

H 9.9F

C

6

8GA B

ED

A B

C

Chapter TestCHAPTER

6


1. Sketch a concave pentagon.

Find the value of each variable.

2. 3. 4. 5.

Decide if you are given enough information to prove that the

quadrilateral is a parallelogram.

6. Diagonals are congruent. 7. Consecutive angles are supplementary.

8. Two pairs of consecutive angles are congruent. 9. The diagonals have the same midpoint.

Decide whether the statement is always, sometimes, or never true.

10. A rectangle is a square. 11. A parallelogram is a trapezoid. 12. A rhombus is a parallelogram.

What special type of quadrilateral is shown? Justify your answer.

13. 14. 15. 16.

17. Refer to the coordinate diagram at the right. Use the Distance Formula to prove that WXYZ is a rhombus. Then explain how the diagram can be used to show that the diagonals of a rhombus bisect each other and areperpendicular.

18. Sketch a kite and label it ABCD. Mark all congruent sides and angles of the kite. State what you know about the diagonals AC

Æand BD

Æand

justify your answer.

19. PLANT STAND You want to build a plant stand with threeequally spaced circular shelves. You want the top shelf to have adiameter of 6 inches and the bottom shelf to have a diameter of15 inches. The diagram at the right shows a vertical cross sectionof the plant stand. What is the diameter of the middle shelf?

20. HIP ROOF The sides of a hip roof form two trapezoids and two triangles, as shown. The two sides not shown arecongruent to the corresponding sides that are shown. Find the total area of the sides of the roof.

66

66

24

24

1012

9

129

9

11

19

x � 62y

7 10x �y �

110�

5x � 6

3x

412 y

100�

70� 75�

x �


y

xY (�a, 0)

X (0, b)

Z (0, �b)

W (a, 0)

32 ft

20 ft

22 ft

17 ft

15 ft

6 in.

15 in.

x in.

Chapter TestCHAPTER

7

35A Chapter 7 Geometry English–Spanish Reviews

In Exercises 1–4, use the diagram.

1. Identify the transformation ¤RST ˘ ¤XYZ.

2. Is RTÆ

congruent to XZÆ

?

3. What is the image of T?

4. What is the preimage of Y?

5. Sketch a polygon that has line symmetry, but not rotational symmetry.

6. Sketch a polygon that has rotational symmetry, but not line symmetry.

Use the diagram, in which lines m and n are lines of reflection.

7. Identify the transformation that maps figure T onto figure T§.

8. Identify the transformation that maps figure T onto figure Tfl.

9. If the measure of the acute angle between m and n is 85°, what isthe angle of rotation from figure T to figure Tfl?

In Exercises 10–12, use the diagram, in which k ∞ m.

10. Identify the transformation that maps figure R onto figure R§.

11. Identify the transformation that maps figure R onto figure Rfl.

12. If the distance between k and m is 5 units, what is the distance between corresponding parts of figure R and figure Rfl?

13. What type of transformation is a composition of a translationfollowed by a reflection in a line parallel to the translation vector?

Give an example of the described composition of transformations.

14. The order in which two transformations are performed affects thefinal image.

15. The order in which two transformations are performed does notaffect the final image.

FLAGS Identify any symmetry in the flag.

16. Switzerland 17. Jamaica 18. United Kingdom

Name all of the isometries that map the frieze pattern onto itself.

19. 20. 21.


Sy

x

T

Y

Z RX1

2

T

n

m

T'T"

R

k m

R' R"

Chapter TestCHAPTER

8


In Exercises 1–3, solve the proportion.

1. �3x

� = �192� 2. �

1y8� = �

12

50� 3. �1

1110� = �1

z0�

Complete the sentence.

4. If �52� = �

ab�, then �

5a� = �b

?�. 5. If �

8x� = �

3y�, then �8 +

xx

� = �?y�.

In Exercises 6–8, use the figure shown.

6. Find the length of EFÆ

.

7. Find the length of FGÆ

.

8. Is quadrilateral FECB similar to quadrilateral GFBA? If so, what is the scale factor?

In Exercises 9–12, use the figure shown.

9. Prove that ¤RSQ ~ ¤RQT.

10. What is the scale factor of ¤RSQ to ¤RQT?

11. Is ¤RSQ similar to ¤QST? Explain.

12. Find the length of QSÆ

.

In Exercises 13–15, use the figure shown to decide if you are given enough

information to conclude that JKÆ

∞ LMÆ

. If so, state the reason.

13. �JLHJ� = �MKH

K�

14. ™HJK £ ™HLM

15. �LJHH� = �MKH

H�

16. The triangle ¤RST is mapped onto ¤R§S§T§ by a dilation with RS = 24,ST = 12, RT = 20, and R§S§ = 6. Find the scale factor k, and side lengthsS§T§ and R§T§.

17. Two sides of a triangle have lengths of 14 inches and 18 inches. Themeasure of the angle included by the sides is 45°. Two sides of a secondtriangle have lengths of 7 inches and 8 inches. The measure of the angleincluded by the sides is 45°. Are the two triangles similar? Explain.

18. You shine a flashlight on a book that is 9 inches tall and 6 inches wide. It makes a shadow on the wall that is 3 feet tall and 2 feet wide. What is the scale factor of the book to its shadow?


A1.4 4.22.8

E

B

F

C

G

D1.5 2.253.2

4.5

25

20

15

R

q T

S

H

L

J

MK

Not drawn to scale

9 in. 3 ft9 in.

Chapter TestCHAPTER

9


Use the diagram at the right to match the angle or segment with its

measure. (Some measures are rounded to two decimal places.)

1. ABÆ

A. 5.33

2. BCÆ

B. 36.87°

3. ADÆ

C. 5

4. ™BAC D. 53.13°

5. ™CAD E. 6.67

6. Refer to the diagram above. Complete the following statement: ¤ABC ~ ¤��? �� ~ ¤��? ��.

7. Classify quadrilateral WXYZ in the diagram at the right. Explain yourreasoning.

8. The vertices of ¤PQR are P(º2, 3), Q(3, 1), and R(0, º3). Decide whether ¤PQR is right, acute, or obtuse.

9. Complete the following statement: 15, ��? , and 113 form a Pythagorean triple.

10. The measure of one angle of a rhombus is 60°. The perimeter of the rhombusis 24 inches. Sketch the rhombus and give its side lengths. Then find its area.

Solve the right triangle. Round decimals to the nearest tenth.

11. 12. 13.

14. L = (3, 7) and M = (7, 4) are the initial and the terminal points of LMÆ„

.Draw LM

Æ„in a coordinate plane. Write the component form of the vector.

Then find its magnitude and direction relative to east.

15. Find the lengths of CDÆ

and ABÆ

. 16. Find the measure of ™BCA and the length of DE

Æ.

Let u„ = ⟨0, º5⟩, v„ = ⟨º2, º3⟩, and w„ = ⟨4, 6⟩. Find the given sum.

17. u„ + v„ 18. u„ + „w 19. v„ + „w

A E

B

35�40

D

C

A BD

C

40�

10

R

4

P

q

6

FD

12

E

25�

LJ

9

K

30�


3 DB

4

A

C

Z

W

Y

8

615 X

17

Chapter TestCHAPTER

10


Use the diagram at the right.

1. Which theorems allow you to conclude that JKÆ

£ MKÆ

?

2. Find the lengths of JKÆ

, MPÆ

, and PKÆ

.

3. Show that JL� £ LM�.

4. Find the measures of JM� and JN�.

Use the diagram at the right.

5. Show that AF� £ AB� and FHÆ

£ BHÆ

.

6. Show that FE� £ BC�.

7. Suppose you were given that PH = PG. What could you conclude?

Find the measure of each numbered angle in ›P.

8. 9. 10. 11.

12. Sketch a pentagon ABCDE inscribed in a circle. Describe the relationshipbetween (a) ™CDE and ™CAE and (b) ™CBE and ™CAE.

In the diagram at the right CAÆ

is tangent to the circle at A.

13. If AG = 2, GD = 9, and BG = 3, find GF.

14. If CF = 12, CB = 3, and CD = 9, find CE.

15. If BF = 9 and CB = 3, find CA.

16. Graph the circle with equation (x º 4)2 + (y + 6)2 = 64.

17. Sketch and describe the locus of points in the coordinate plane that areequidistant from (0, 3) and (3, 0) and 4 units from the point (4, 0).

18. ROCK CIRCLE This circle of rock is in the Ténéré desert in the African country of Niger. The circle is about 60 feet in diameter. About a mile away to the north, south, east, and west, stone arrows point away from the circle. It’s not known who created the circle or why. Suppose the center of the circle is at (30, 30) on a grid measured in units of feet. Write an equation for the circle.

19. DOG RUN A dog on a leash is able to move freely along a cable that is attached to the ground. The leash allows the dog to move anywhere within 3.5 feet from any point on the 10-foot straight cable. Draw and describe the locus of points that the dog can reach.

P2

36�1

96�3

38�P 2

105�1

60�P

3

2

1P

1

2

145�


MK

P

N1

J

4 4

H

L

A

BC

D

E

FG

A B

C

DE

F

P

H

G

Chapter TestCHAPTER

11


In Exercises 1 and 2, use the figure at the right.

1. What is the value of x?

2. Find the sum of the measures of the exterior angles, one at each vertex.

3. What is the measure of each interior angle of a regular 30-gon?

4. What is the measure of each exterior angle of a regular 27-gon?

In Exercises 5º8, find the area of the regular polygon to two decimal places.

5. An equilateral triangle with perimeter 30 feet 6. A regular pentagon with apothem 8 inches

7. A regular hexagon with 9 centimeter sides 8. A regular nonagon (9-gon) with radius 1 meter

Rhombus ABCD has sides of length 8 centimeters. EFGH is a similar rhombus with

sides of length 6 centimeters.

9. Find the ratio of the perimeters of ABCD to EFGH. Then find the ratio oftheir areas.

10. The area of ABCD is 56 square centimeters. Find the area of EFGH.

Use the diagram of ›R.

11. Find the circumference and the area of ›R.

12. Find the length of AB�.

13. Find the area of the sector ARB.

Find the area of the shaded region.

14. 15. 16.

In Exercises 17 and 18, a point is chosen randomly in the 20 inch by

20 inch square at the right.

17. Find the probability that the point is inside the circle.

18. Find the probability that the point is in the shaded area.

19. WATER-SKIER A boat that is pulling a water-skier drives in a circlethat has a radius of 80 feet. The skier is moving outside the path of the boat in a circle that has a radius of 110 feet. Find the distance traveled by the boat when it has completed a full circle. How much farther has the skier traveled?

20. WAITING TIME You are expecting friends to come by your house any time between 6:00 P.M. and 8:00 P.M. Meanwhile, a problem at work has delayed you. If you get home at 6:20 P.M., what is the probability that your friends are already there?

120�

7 m

120�

7 m

115�16 in.

30 ft


120�

135�

90�

115�120�

x �

105�

A

B

R

5 cm

20 in.

20 in.

Chapter TestCHAPTER

12


Determine the number of faces, vertices, and edges of the solids.

1. 2. 3.

USING ALGEBRA Sketch the solid described and find its missing

measurement. (B is the base area, P is the base perimeter, h is the height,

S is the surface area, r is the radius, and l is the slant height.)

4. Right rectangular prism: B = 44 m2, P = 30 m, h = 7 m, S = ��?�

5. Right cylinder: r = 8.6 in., h = ��?� , S = 784π in.2

6. Regular pyramid: B = 100 ft2, P = 40 ft, l = ��?� , S = 340 ft2

7. Right cone: r = 12 yd, l = 17 yd, S = ��?�

8. Sphere: r = 34 cm, S = ��?�

In Exercises 9–11, find the volume of the right solid.

9. 10. 11.

12. Draw a net for each solid in Exercises 9–11. Label the dimensions of thenet.

13. The scale factor of two spheres is 1:5. The radius of the smaller sphere is 3 centimeters. What is the volume of the larger sphere?

14. Describe the possible intersections of a plane and a sphere.

15. What is the scale factor of the two cylinders at the right?

16. CANNED GOODS Find the volume and surface area of a prism with aheight of 6 inches and a 4 inch by 4 inch square base. Compare the resultswith the volume and surface area of a cylinder with a height of 7.64 inchesand a diameter of 4 inches.

SILOS Suppose you are building a silo. The shape of your silo is a

right prism with a regular 15-gon for a base, as shown. The height of

your silo is 59 feet.

17. What is the area of the floor of your silo?

18. Find the lateral area and volume of your silo.

19. What are the lateral area and volume of a larger silo that is in a 1:1.25 ratio with yours?

5 ft7 ft

6 ft

12 cm

15 ft

20 ft

18 ft

xyxy


4 ft

V = 8π m3 V = 27π m3

Answer Key

Chapter 1 Test (pages 5A–5B)

1. Sample answer: 2. Sample answer

Q, T, and N Q, N, M, and R

3. Sample answer: 4. Sample answer:

and and

5. 6. 7.

8. 9.

10.

11.

12.

13.

14. Sample answer:

is an obtuse angle. is an acute angle.is right angle. and are complementa-

ry angles.

15.

� 15

� 4 � 11

QR � 4 � w

w � 11

� 19 3w � 33

� 22 � 3 3w � 1 � 34

� 2 � 11 � 3 2w � 3 � 4 � w � 34

PQ � 2w � 3 PQ � QR � PR

�EBD�FBE�FBD�DBC�ABD

m�DBA � 130�

m�DBA � 50� � 180�

m�DBA � m�DBC � 180�

m�ABF � 40�

m�ABF � 140� � 180�

m�ABF � m�FBC � 180�

� 140�

m�FBC � 45� � 45� � 50�

m�FBC � m�FBE � m�EBD � m�DBC

m�DBE � 45�

m�DBE � 45� � 90�

m�DBE � m�EBF � m�DBF

NR � 14

� 22 12 � NR � 26

� 8 � 14 4 � 8 � NR � 26

MR � MN � NRSM � MN � NR � SR

SM � MP � 4� 4�12 �8�MP �

12 �MN�

´QL

´LQ

´QN

ÆTN

ÆTQ

16.

The coordinates of point S are

So

17.

18.

19.

20. The distance is 6 more than twice the figure number. Forthe 20th figure, the distance is 2(20) � 6.

2(20) � 6 � 40 � 6

� 46

The distance around the 20th figure is 46 units.

21.

They would have walked about 3517 feet.

22.

The area of the watered region is about 984,704 squarefeet.

� 984,704

� 3.14 � 313,600

� 3.14�560�2

A � �r2

� 3517

� 2 � 3.14 � 560

C � 2�r

� 65�

� 12 � 130�

m�PQT �12�m�PQR�

m�5 � m�3 � 68�

m�4 � 112�

m�4 � 68� � 180

m�4 � m�3 � 180�

RS � ST.

� �10

� �9 � 1

� �32 � ��1�2

ST � ��3 � 0�2 � �6 � 7�2

� �10

� �9 � 1

� �32 � ��1�2

RS � ��0 � ��3��2 � �7 � 8�2

�0, 7�.

� �0, 7�

� �02

, 142

S � ��3 � 32

, 8 � 6

2

figure 1 2 3 4 5

distance 8 10 12 14 16

Answer Key


1. Postulate 9: A plane contains at least three noncollinearpoints.

2. Postulate 11: If two planes intersect, then their intersectionis a line.

3. Postulate 8: Through any three noncollinear points thereexists exactly one plane.

4. Postulate 10: If two points lie in a plane, then the line con-taining them lies in the plane.

5.–8. Sample answers are given

5. A angle is acute, but it does not measure

6. Two segments of length 10 feet each are congruent, buttheir lengths do not measure 17 feet.

7. Two angles of measure and are supplementary,but their measures do not equal and

8. You can choose the prime number 7, which is not equal to13.

9. 10. 11. 12.

13. Symmetric property of equality

14. Multiplication property of equality

15. Transitive property of equality

16. Subtraction property of equality

17. Statements Reasons

1. 1. Given

2. 2. Addition Postulate ofEquality

3. 3. Segment Addition Postulate

4. 4. Transitive property ofequality

5. 5. Definition of congruentsegments

18. Transitive property of segment congruence: If and , then . Let represent thesection of leaking pipe, the section of the measuringtape, and the new section of pipe.

19. �1 �4 by the Congruent Complements Theorem.

EFCD

ABEFABEFCDCDAB

AC � DB

AC � DB

DX � XB � DBAX � XC � AC

AX � XC � DX � XB

XB � XCAX � DX

z � 30y � �5y � 17y � 70

148�.32�50�130�

34�.60�

Answer Key


1. alternate interior angles 2. supplementary angles

3. corresponding angles 4. alternate exterior angles

5. consecutive interior angles 6. vertical angles

7.

8. 9. 10. 11.

12. 13.

14. Sample answer: Washington Monument; yes. The lines

that form the top are skew to the opposite sides of the base

of the monument.

15. Yes; Theorem 3.12: If two coplanar lines are perpendicu-lar to the same line, then they are parallel to each other.

y �12

x �52

52

� b y � �13

x � 1

2 � �12

� b 1 � b

2 �12

��1� � b 1 � �13

�0� � b

p � q; l � mm � nl � n�2, �4, �6, �8

16. Statements Reasons

1. 1. Given

2. 2. Corresponding anglesconverse

3. 3. Alternate interior anglestheorem

4. 4. Transitive propertyof angle congruence

5. 5. Corresponding anglesconverse

n �� p

�3 � �5

�4 � �5

l � m

�1 � �2, �3 � �4

m�1 � m�3 � 37°

m�ABC � 90°

m�1 � m�2 � m�3� m�ABC

fiBA�

fiBC

�ABC is a right angle

m�1 � m�2 � m�3 � 90�

37� � m�2 � 37� � 90�

m�2 � 16�

Given

Angle addition postulate

Transitive property of equality

Substitution property of equality

Subtraction property of equality

Definition of lines�

Definition of a rightangle

Given

Answer Key


1. 2. 3. 4.

5. 6.

7.

Let

Therefore

8. Yes; AAS Congruence Theorem; two pairs of correspon-ding angles are congruent and two corresponding nonin-cluded sides are congruent.

9. Yes; SAS Congruence Postulate; (given), and(Reflexive Property of congruence). Since, (Alternate Interior Angles

Theorem), so two pairs of corresponding sides are congruent and two corresponding included angles are con-gruent.

10. Yes; ASA Congruence Postulate; since and (given), and (ReflexiveProperty of Congruence), two pairs of correspondingangles are congruent and two corresponding includedangles are congruent.

11. No; the triangles cannot be proved congruent from thegiven information.

12. Yes; HL Congruence Theorem; since (given)and (Reflexive Property of Congruence), twocorresponding hypotenuses and two corresponding legs oftwo right triangles are congruent.

13. Yes; HL Congruence Theorem; since it is given thatand are right triangles and and

two corresponding hypotenuses and two corre-sponding legs of two right triangles are congruent.

14.

15.

16.

x � 50

x � 130 � 180

x� � 2�65�� 180�

x � 5

3x � 4 � 2x � 1

x � 55

2x � 110

70� � x� � x� � 180�

KM � MH,HG � KJ�LMK�GMH

XZ � XZWX � XY

MP � MP�NMP � �LPM�LMP � �NPM

�HJG � �JGKHJ � GKGJ � GJ

HJ � KG

m�C � 16� and m�B � 3�16�� 48�

x � 16

4x � 64

116� � 3x� � x� � 180�

x� � m�C

m�A � m�B � m�C � 180�

�QPR�QSR

�QPS�QPR�QPS�QPS, �QSR

17.

Statements Reasons

1. 1. Given

2. 2. Base Angles Theorem

3. 3. SAS Congruence Postulate

4. 4. Corresp. parts of s are

18.

Statements Reasons

1. 1. Given

2. 2. Alternate Interior AnglesTheorem

3. 3. Reflexive Property of Congruence

4. 4. SAS Congruence Postulate

5. 5. Corresp. parts of s are

19. Using the Pythagorean Theorem,

20. M � �s � 02

, s � 0

2 � �s2

, s2

�42 � 72 � �16 � 49 � �65.

� . � ��X � �W

�XYZ � �WZY

ZY � ZY

�XZY � �WYZ�XYZ � �WZY,

XY � WZ, XZ � WY

� . � �AB � AE

�ABD � �AEC

�1 � �2

BD � EC, AC � AD

Answer Key


1. If P is the circumcenter of then PR, PS, and PTare always equal.

2. If bisects then and are sometimes con-gruent.

3. The incenter of a triangle never lies outside the triangle.

4. The length of a median of a triangle is sometimes equal tothe length of a midsegment.

5. If is the altitude to side of then isalways shorter than

6. a.

b.

c.

d.

7. Point H is the centroid of the triangle.

8. is a(n) median, perpendicular bisector, altitude, andangle bisector of

9. and by the Midsegment Theorem.

10. because the side opposite islonger than the side opposite

11. To locate the pool so that its center is equidistant from thesidewalks, find the incenter of the triangle by constructingangle bisectors of two angles of the triangle and locatingthe point of intersection of the bisectors. This point will beequidistant from each sidewalk.

12. The converse of the Hinge Theorem guarantees that theangles between the legs get larger as the legs are spreadapart.

�ACB.�BACm�BAC > m�ACB

EF � ABEF �12 AB

�ABC.CG

BC � 19.8

BC � 9.9 � 9.9

BC � CF � FB

HE � 5

23 HE �103

HE �13 HE �

103

HE �13 �HE � 10�

HE �13 �HE � HB�

HE �13 EB

10 � HB

100 � �HB�2

36 � 64 � �HB�2

62 � 82 � �HB�2

�HG�2 � �GB�2 � �HB�2

HC � 12

13 HC = 4

HC �23 HC � 4

HC �23 �HC � 6�

HC �23 �HC � HG�

HC �23 CG

AB.AM�ABC,BCAM

CDAD�ABC,ÆBD

�RST,

13. The maximum distance between the end of two legs is 10 feet because the length of the third side of the trianglemust be less than the sum of the lengths of the other twosides.

14.

If then is longer than because two sides of one triangle are congruent to twosides in another triangle and the measure of the includedangle of one triangle is larger than the measure of theincluded angle of the other triangle (Hinge Thm.).

15.

Statements Reasons

1. 1. Given

2. 2. Segment Addition Post.

3. 3. Substitution property of equality

4. 4. Triangle Inequality Theorem

5. 5. Substitution property ofequality

16. Assume Then because if twoangles of a triangle are congruent, then the sides oppositethem are congruent. So by the definition of con-gruent segments. But this contradicts the given statementthat Therefore, the assumption must be false.So m�D � m�ABC.

AD � AB.

AD � AB

AD � ABm�D � m�ABC.

BE < AE

BE < BC � CE

BC � CE � AE

AC � CE � AE

AC � BC

BCACm�AOC > m�BOC,

A BC

O

Answer Key


1. Sample answer:

2.

3.

4.

y � 180 � 110 � 70

5. 6. no 7. yes 8. no

9. yes 10. sometimes 11. never 12. always

13. Trapezoid; exactly one pair of sides are parallel.

14. Rhombus; the diagonals bisect each other and are perpendicular.

15. Rectangle; one pair of opposite sides are both and ,and since one angle is a right angle, the opposite angle andthe two consecutive angles are also right angles.

16. Square; the diagonals bisect each other and are congruentso the quad. is a �, a rectangle, and a rhombus. So it is asquare.

17.

So,

Let O be the origin (where the diagonals meet). OX � OZ� b and OW � OY � a, so the diagonals bisect each other.One diagonal is vertical and the other is horizontal, so theyare perpendicular.

WX � XY � YZ � ZW.

� �a2 � b2 ZW � ��0 � a�2 � ��b � 0�2

� �a2 � b2 YZ � ��0 � ��a��2 � ��b � 0�2

� �a2 � b2 XY � ��0 � ��a��2 � �b � 0�2

� �a2 � b2 WX � ��a � 0�2 � �0 � b�2

�

x � 4 y �72

x � 6 � 10 2y � 7

x � 110

x � 3

y � 8 �2x � �6

12y � 4 3x � 5x � 6

x � 115

x � 245 � 360

x� � 100� � 70� � 75� � 360�

18. 19.

; �ABC �ADC (SSS Cong. Post.) so �BAC �DAC. Then �BAP �DAP (SAS Cong.Post.). Corr. � BPA and DPA are . Since and form a linear pair of �, .

20.

� 1218 ft2 � 918 � 300

� �32 � 22��17� � �20��15� A � 2�1

2�b1 � b2�h� � 2�12bh�

AC � BDBDAC

AC � BD

� 10.5 in.

x �12�6 � 15�

D

B

CA P

Answer Key


1. The transformation that maps onto is areflection in the y-axis.

2. Yes, because a reflection preserves length.

3. The image of T is Z. 4. The preimage of Y is S.

5. Sample answer: 6. Sample answer:

7. The transformation that maps figure T onto figure is areflection in line m.

8. The transformation that maps figure T onto is a reflec-tion in line m followed by a reflection in line n. Or it is arotation about the point of intersection of lines m and n.

9. The measure of the angle of rotation is twice the measureof the acute angle formed by lines m and n. So, the meas-ure of the angle of rotation is or

10. The transformation that maps figure R onto figure is areflection in line k.

11. The transformation that maps figure R onto is a reflec-tion in line k, followed by a reflection in line m, or it is atranslation.

12. The distance between corresponding parts of figure R andfigure is twice the distance between lines k and m. Thedistance is or 10 units.

13. A glide reflection is a composition of a translation fol-lowed by a reflection in a line parallel to the translationvector.

14. Sample answer:

is the final image when is rotated clockwiseabout the origin, then reflected in the y-axis. is thefinal image when is reflected in the y-axis, then rotated clockwise about the origin.90�

ABAB

90�ABA�B�

B

A

B�

A�B

A

y

x

1

1�1�1

2 � 5R

R

R�

170�.2 � 85�

T

T�

RT � XZ

�XYZ�RST

15. Sample answer:

is reflected in the x-axis, then translatedThe same image results if the

transformations are performed in reverse order.

16. The flag of Switzerland has a vertical line of symmetry,a horizontal line of symmetry, two diagonal lines of symmetry and rotational symmetry. It can be mapped ontoitself by a clockwise or counterclockwise rotation of about the center.

17. The flag of Jamaica has a vertical line of symmetry, a hori-zontal line of symmetry, and rotational symmetry. It can bemapped onto itself by a clockwise or counterclockwiserotation of about the center.

18. The flag of the United Kingdom has a vertical line of sym-metry, a horizontal line of symmetry, and rotational sym-metry. It can be mapped onto itself by a clockwise or counterclockwise rotation of about its center.

19. translation, 180° rotation, horizontal line reflection,vertical line reflection, glide reflection

20. translation, 180° rotation, horizontal line reflection,vertical line reflection, glide reflection

21. translation, vertical line rotation

180�

180�

180�

�x, y�fi �x � 5, y�.PQ

y

x

1

1

Q(4, 6)

Q�(9, �6)

P(1, 2)

P�(6, �2)

Answer Key


1. 2.

3. 4. 2

5. 6.

7. 8. No;

9. since all right are Since by the Reflexive prop. of cong.,

by the AA Similarity Post.

10.

11. Yes; Sample answer: Since is a right isalso a right Also, since corresp.

are Then by the AASimilarity Post.

12.

13. Yes; Proportionality Converse

14. Yes; Corresponding Converse

15. Yes; Sample answer: by SAS SimilarityThm., def. of s , and Corres. Converse.

16.

17. No;

18. k �9 in.3 ft

�9 in.

36 in.�

14

78

�1418

R�T� � 5 S�T� � 3

4�R�T�� 20 4�S�T�� 12

14

�R�T�

2014

�S�T�

12k �

624

�14

��~�LHM ~ �JHK

�

�

x � 12

5x � 60

35

�x

20

�RSQ ~�QST�.� RQS and T�RSQ ~ �RQT,�.

�QST�,�RSQ

1525

�35

�RSQ ~�RQT �R � �R

�.��RSQ � �RQT

FG � 4.8

2.8�FG� � 13.44

1.52.25

�1.44.2

2.83.2

�4.2FG

EF � 1.6

2.8�EF� � 4.48

2.83.2

�1.4EF

3 � y

z � 1

110z � 110

11110

�z

10

y � 24 x � 4

15y � 360 9x � 36

18y

�1520

x3

�129

Answer Key


1. E 2. A 3. C 4. D 5. B

6.

7. is a kite. The diagonals are perpendicular and thequadrilateral has two pairs of consecutive congruent sides,but opposite sides are not congruent.

8. 9.

acute

10.

side length

11.

12.

13.

14.

15.

16.

17. 18. 19. 2, 3� 4, 1� �2, �8�

DE � 32.7 BC � 22.9

tan 35� �22.9DE

sin 35� �BC40

m�BCA � 90� � 35� � 55�

AB � 13.1 CD � 6.4

sin 50� �10AB

sin 40� �CD10

x � 36.9� south of east

tan x �34

�LM\

� � �16 � 9 � 5

LM\

: 4, �3�y

x1

1

L

M

� 4.5

m�R � 41.8� m�P � 48.2� QR � 2�5

sin R �46

cos P �46

42 � QR2 � 62

m�F � 90� � 25� � 65�

DE � 25.7 DF � 28.4

tan 25� �12DE

sin 25� �12DF

m�K � 90� � 30� � 60�

JL � 7.8 KL � 4.5

cos 30� �JL9

sin 30� �KL9

� 6 in.

� 31.2 in.2� 18�3

�12

�6��6�3 �

A �12

d1d2

6 in.

6 in.6 in.

30�

3 3 in. 3 in.

60�

6 in.

40 < 54

�2�10 �2 ? ��29 �2 � 52

b � 112PR � �4 � 36 � 2�10

b2 � 12,544QR � �9 � 16 � 5

152 � b2 � 1132PQ � �25 � 4 � �29

WXYZ

DBA; DAC

Answer Key


1. Thm. 10.3; HL Congruence Thm.

2.

3. and are rt because tangents are toradii at the points of tangency. and are radii of so they are by the Reflexive Prop. ofCongruence. So by the HL Cong. Thm.

because they are corresponding angles.By the defs. of congruence, minor arcs, and congruentarcs,

4. is a short leg: 4; long leg:hypotenuse: 8.

m∠ MHP �

5. According to Thm. 10.5, since and

6. so by Thm. 10.4

7. and

8. 9.

10. 11.

12. a. supplementary

b. congruent

13. 14. 15.

x � 6 x � 4 x � 6

x2 � 3�12� 3�12� � x�9� 3x � 18

m�CBE � m�CAE

m�CDE � 2m�CAE

E

D

C

B

A

m�3 � 37�

m�2 � 66�m�2 � 75�

m�1 � 29�m�1 � 120�

m�3 � 45�

m�2 � 90�m�2 � 145�

m�1 � 90�m�1 � 72.5�

FB� � EC�FB � EC

FE� � BC� .FE � BC

� BA� .FA�FH � BHAD � FB,

m�PHJ � m�MHJ � m�MHP � 156.9� � mJN�tan�1 PM

HM � tan�1 34 � 36.9�

m�MHJ � 120� � mJM�m�MHK � 60� � m�JHK

4�3;30� � 60� � 90��,�HMK

LM� � JL� .

�KHM � �KHJ�KJH � �KMH

HK � HK� .�HHMHJ

��s�KMH�KJH

PK � 4�3 � 3

MK � 4�3MP � 3

�MK�2 � 48�MP�2 � 9

�MK�2 � 42 � 82�MP�2 � 42 � 52

JK � 4�3 � 6.9

42 � �JK�2 � 82

16. 17.

Two points, (0, 0) and (4, 4), where the line and circle with center and radius 4 units intersect

18.

19.

Let A and B be the ends of the cable. The locus consists ofthe points on or inside a region bounded by two semicircleswith centers A and B and radius 3.5 ft and two segments onopp. sides of to and 3.5 ft from AB.ABAB, both �

A B

3.5 ft

3.5 ft

�x � 30�2 � �y � 30�2 � 900

�4, 0�

y � x�x � 4�2 � �y � 6�2 � 64

y

x

1

1

y

x

2

�22

Answer Key


1.

2.

3. measure of interior

4. meas. of exterior 5.

6.

7.

8.

9. 10.

11.

� 78.54 cm2 � 31.42 cm

� � �5�2 � 2� �5�

A � �r2 C � 2�r

x � 31.5 cm2

16x � 504Area of ABCDArea of EFGH

�42

32 �169

56x

�169

Perimeter ABCDPerimeter EFGH

�86

�43

� 2.89 m2

�12

�cos 70��9��2�� sin 70��

A �12

aP

Measure of interior angle ��9 � 2��180��

9� 140�

� 210.44 cm2

A � 6 �14�3�9�2

� 232.49 in.2

�12

�8��5��2� 8tan 54�

A �12

aP

Measure of interior angle ��5 � 2��180��

5� 108�

� 43.30 ft2 �

14�3�10�2� 13

13

�

A �14�3s2� �

360�

27

� 168�

��30 � 2��180��

30

� ��n � 2��180��

n

40� � 60� � 45� � 90� � 65� � 60� � 360�

x� � 140�

x� � 580� � 720�

12. Length of

13.

14.

15.

16.

17. 18.

19.

20. P �20 min

120 min�

16

� 16.67%

691.15 � 502.65 � 188.50 ft

skiier � 2� �110� � 691.15 ft

boat � 2� �80� � 502.65 ft

�18

or 12.5%

�50

400 � 0.785 or 78.5%

�12�10��10�

202 �� 10�2

�20�2

P �Area of triangleArea of square

P �Area of circleArea of square

� 51.31 m2

Area of sectors �120�

360�� 7�2

� 547.34 in.2

Area of sector �245�

360�� 16�2

� 353.43 ft2

�12

� �15�2

A �12

�r2

� 22.91 cm2

�105�

360�� 5�2

Area sector ARB �mAB�360�

� �r2

� 9.16 cm

�105�

360�� 2� �5�

�m AB�360�

2�rAB� x� � 120� � 135� � 90� � 115� � 120� � �6 � 2��180��

Answer Key


1. 2.

3.

4.

5.

6.

7.

8.

9. 10.

11.

� 70 ft3 � 1

3�7��5��6� V �

13Bh

� 2375.04 cm3 � 2700 ft 3

� �3.14��6�2�21� �12�20��15��18�

V � �r2h V � Bh

� 14,526.72 cm2

� 4�3.14��34�2

S � 4�r2

34 cm

� 1093.27 yd2

S � �3.14��12�2 � �3.14��12��17� S � �r2 � �rl

12 yd

17 yd

12 ft � l

240 � 20l

340 � 100 �12�40�l

S � B �12Pl

100 ft2

36.98 in. � h

636.08� � 17.2�h

784� � 147.92� � 17.2�h

784� � 2��8.6�2 � 2��8.6�h S � 2�r2 � 2�rh8.6 in.

� 298 m2

� 2�44� � �30��7� S � 2B � Ph

44 m2

7 m

F � 9; V � 14; E � 21

F � 5; V � 6; E � 9F � 8; V � 12; E � 18

12.

13.

14. A plane may intersect a sphere in a point or in a circle. Ifthe plane contains a diameter of the sphere, the intersectionis a great circle.

15.

16. Prism Cylinder

The surface area of the cylinder is less than that of theprism, but the volumes are the same.

� 96.01 in.3 � 96 in.3� �3.14��2�2�7.64� � �16��6�

V � �r2h V � Bh

� 121.14 in.2 � 128 in.2� 2�3.14��2�2 � 2�3.14��2��7.64� � 2�16� � �16��6�

S � 2�r2 � 2�rh S � 2B � Ph

3�83�27

�23

� 14,137.17 cm3

�43

��15�3

V �43

�3

r � 5 � 3 cm � 15 cm

7 ft5 ft

6.5 ft

6.95 ft6.95 ft

6.5 ft

21 cm

12 cm

12 cm

18 ft

25 ft15 ft

20 ft20 ft

20 ft

Answer Key

17. 18.

19.

� 32,496.09 ft3 V � �1.25�3�16,638�

V: � 11.25

3

�1

1.95

� 5531.25 ft2 Lateral Area � �1.5625��3540�

A: � 11.25

2

�1

1.5625

� 16,638 ft3 � �282��59�

V � Bh

� 3540 ft2 � 282 ft2 � �15��4��59� � 1

2�9.4��15��4� L.A. � Ph A � 12aP

geometry chapter tests with key

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